Vol. 7, No. 1, 2014

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Comparing a series to an integral

Leon Siegel

Vol. 7 (2014), No. 1, 57–65

We consider the difference between the definite integral 0uxeu du, where x is a real parameter, and the approximating sum k=1kxek. We use properties of Bernoulli numbers to show that this difference is unbounded and has infinitely many zeros. We also conjecture that the sign of the difference at any positive integer n is determined by the sign of cos((n + 1)arctan(2π)).

polylogarithms, gamma function, Bernoulli numbers
Mathematical Subject Classification 2010
Primary: 33B15
Received: 17 July 2012
Revised: 25 May 2013
Accepted: 25 May 2013
Published: 24 October 2013

Communicated by Andrew Granville
Leon Siegel
Christian-Albrechts-Universität zu Kiel
Christian-Albrechts-Platz 4
24118 Kiel